Realistic Bell tests with homodyne measurements

We analyze Bell inequalities violations in photonic experiments for which the measurement apparatuses are restricted to homodyne measurements. Through numerical optimization of the Clauser-Horne-Shimony-Holt inequality over homodyne measurements and binning choices, we demonstrate large violations for states with a bounded number of photons. When considering states defined within qubit local subspaces of two Fock states, such as NOON states, a violation is observed solely within the qubit Fock space spanned by zero and two photons. For more generic states, large violations are obtained. Significant violations are observed even for states containing three photons locally and under realistic values of noise and losses. We propose concrete implementations to achieve such violations, opening new avenues for Bell experiments with homodyne detectors.